We present a study of the collisional relaxation of ion velocities in a strongly coupled, ultracold neutral plasma on short time scales compared to the inverse collision rate. The measured average velocity of a tagged population of ions is shown to be equivalent to the ion-velocity autocorrelation function. We thus gain access to fundamental aspects of the single-particle dynamics in strongly coupled plasmas and to the ion self-diffusion constant under conditions where experimental measurements have been lacking. Nonexponential decay towards equilibrium of the average velocity heralds non-Markovian dynamics that are not predicted by traditional descriptions of weakly coupled plasmas. This demonstrates the utility of ultracold neutral plasmas for studying the effects of strong coupling on collisional processes, which is of interest for dense laboratory and astrophysical plasmas.